Special Triangles: 30-60-90 and 45-45-90

Special Triangles: 30-60-90 and 45-45-90

You have definitely heard of acute triangles, obtuse triangles, or even equilateral triangles, but have you heard of ‘special triangles’? Special Triangles are right angled triangles that have sides and angles which are always in a fixed ratio. There are two types of special triangles: 30-60-90 - pronounced ‘thirty-sixty-ninety’ and 45-45-90 - pronounced ‘forty five-forty five-ninety.’

What is a 30-60-90 triangle?

As the triangle has an angle that measures 90 degrees, it is a right triangle. So, 30-60-90 is a right triangle that always has angles measuring 30 degrees, 60 degrees, and 90 degrees. This triangle also has side length values that are always in a consistent relationship with one another.

If the side length opposite to the 30 degree angle is x,

then, the side length opposite to the 60 degree angle is x√3

and the side length opposite to the 90 degree angle is 2x.

So, in the following triangle

  

Side length opposite to the 30 degree angle is 8

So, the side length opposite to the 60 degree angle is 8√3

and the side length opposite to the 90 degree angle is 16.

Similarly, in the following triangle,

Side length opposite to the 60 degree angle is 12

So, the side length opposite to the 30 degree angle is 4√3

and the side length opposite to the 90 degree angle is 8√3.

And in the following triangle

Side length opposite to the 90 degree angle is 16

So, the side length opposite to the 30 degree angle is 8

and the side length opposite to the 60 degree angle is 8√3.

What is a 45-45-90 triangle?

As the triangle has an angle that measures 90 degrees, it is a right triangle. So, 45-45-90 is a right triangle that always has angles measuring 45 degrees, 45 degrees, and 90 degrees. As there are two equal angles in the triangle, there are two equal sides in this triangle. This triangle also has side length values that are always in a consistent relationship with one another.

If the side length opposite to the 45 degree angle is x,

then, the side length opposite to the other 45 degree angle is also x

and the side length opposite to the 90 degree angle is x√2.

So, in the following triangle

  

Side length opposite to the 45 degree angle is 7

So, the side length opposite to the other 45 degree angle is also 7

and the side length opposite to the 90 degree angle is 7√2.

Similarly, in the following triangle,

Side length opposite to the 90 degree angle is √8 = 2√2

So, the side length opposite to the 45 degree angle is 2

and the side length opposite to the other 45 degree angle is also 2.

Now, let’s try a SAT like question based on the two special triangles.

In the triangle above, what is the length of the side AC?

Solution:

The sum of angles of a triangle is 180.

Sp, angle A + angle B + angle C = 180

=> Angle A + 45 + 105 = 180

=> Angle A = 180 - 150 = 30

Now, drop a perpendicular from point C to side AB.

Now, we have a 45-45-90 triangle and a 30-60-90 triangle.

In a 45-45-90 triangle, the ratio of the sides is x:x:x√2

Here, x√2 = 4√2, so x = 4

Hence, the triangle has the side lengths as follows:

In a 30-60-90 triangle, the ratio of sides is x:x√3:2x

Here, x = 4

So, x√3 = 4√3 and 2x = 8

So, the side lengths of the triangle are as follows:

Hence, the length of the side AC is 8.

Remembering the rules for 30-60-90 triangles and the 45-45-90 triangles will help you to shortcut your way through a variety of math problems.

Keep track of the rules of x, x√3, 2x for 30-60-90 and x, x, x√2 for 45-45-90 triangle in whatever way makes sense to you and try to keep them straight if you can, but don't panic if your mind blanks out when it's crunch time. Either way, you've got this.

Happy prepping!